Proper Affine Vector Fields in Plane Symmetric Static Space-Times Ghulam Shabbir Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, Pakistan Email:
[email protected] and Nisar Ahmed Faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, Pakistan.
Abstract: A study of proper affine vector fields in plane symmetric static spacetimes by using the rank of the 6× 6 Rieman matrix and holonomy. Studying
proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields. 1. INTRODUCTION
Over the recent years, there has been much interest in the study of symmetry in general relativity; affine symmetry seems to ignored probably because the complications involved in (1). Following [1,9] some progresss was made in [2] where they actually shown that under what conditions the space-times admit proper affine vector fields. Other different approaches may be seen in [9,10,11]. In this paper a different approach is devolped to study proper affine vector fields in plane symmetric static space-times by using holonomy, the rank of the 6× 6 Rieman matrix and direct integration techniques. Through out M is representing the four dimensional, connected, hausdorff space-time manifold with Lorentz metric g of signature (-, +, +, +). The curvature tensor associated with g, through Levi-Civita connection, is denoted in component form by R a bcd . The
1
usual covariant, partial and Lie derivatives are denoted by a semicolon, a comma and the symbol L, respectively. Round and square brackets denote the usual symmetrization and skew-symmetrization, respectively. The space-time M will be assumed nonflat in the sense that the Riemann tensor does not vanish over any nonempty open subset of M . A vector field X on M is called an affine vector field if it satisfies X a ;bc = Rabcd X d .
(1)
If one decomposes X a ;b on M into its symmetric and skew-symmetric parts
X a ;b =
1 hab + Fab 2
(hab = hba , Fab = − Fba )
(2)
then equation (1) is equivalent to (i ) hab;c = 0 (ii ) Fab;c = Rabcd X d
(iii ) Fab;c X c = 0.
(3)
The above vector field X is called an affine on vector field M if the local diffeomorphisms ψ t (for appropriate t ) associated with X map geodesics into geodesics and also preserve their affine parameter. If hab = 2cg ab , c ∈ ℜ, then the vector field X is called homothetic (and Killing if c = 0 ). The vector field X is said to be proper affine if it is not homothetic vector field and also X is said to be proper homothetic vector field if it is not Killing vector field on M [1]. It is important to note the following equation
Rabcd k d = 0.
(4)
2. Affine Vector Fields In this section we will briefly discuss when the space-times admit proper affine vector fields for further details see [2,8]. Suppose that M is a simple connected space-time. Then the holonomy group of M is a connected Lie subgroup of the idenity component of the Lorentz group and is thus characterized by its subalgebra in the Lorentz algebra. These have been labeled into fifteen types R1 − R15 [3,4]. It follows from [2] that the only such space-times which could admit proper affine vector fields are those which admit nowhere zero covariantly constant second order symmetric tensor
2
field hab and it is known that this forces the holonomy type to be either R2 , R3 , R4 , R6 , R7 , R8 , R10 , R11 or R13 . Here, we will only discuss the space-times
which has the holonomy type R2 , R4 , R7 , R10 or R13 . First consider the case when M has type R13 . Then one can always set up local coordinates (t , x1 , x 2 , x 3 ) on an open set U = U 1 × U 2 , where U 1 is a one dimensional timelike submanifold of U coordinatized by t and U 2 is a three dimensional spacelike submanifold of U coordinatized by x1 , x 2 , x 3 and where the above product is a metric product and the metric on U is given by [1] ds 2 = −dt 2 + g αβ dx α dx β
(α , β = 1, 2,3)
(5)
where g αβ depends on x γ , (γ = 1, 2, 3). The above space-time is clearly 1+3 decomposable. The curvature rank of the above space-time is atmost three and there exists a unique nowhere zero vector field t a = t ,a satisfying t a;b = 0 and also t a t a = −1. From the Ricci Identity R a bcd t d = 0. It follows from [2] that affine vector fields in this case are X = ( c1t + c 2 )
∂ +Y ∂t
(6)
where c1 , c 2 ∈ ℜ and Y is a homothetic vector field in the induced geometry on each of the three dimensional submanifolds of constant t. Now consider the situation when M has type R10 . The situation is similar to that of previous R13 case except that now we have local decomposition is U = U 1 × U 2 , where U 1 is a one dimensional spacelike submanifold of U and U 2 is a three dimensional timelike submanifold of U . The space-time metric on U is given by [1] ds 2 = dx 2 + g αβ dx α dx β
(α , β = 0, 2,3)
(7)
where g αβ depends on x γ , (γ = 0, 2, 3). The above space-time is clearly 1+3 decomposable. The curvature rank of the above space-time is atmost three and there exists a unique nowhere zero vector field x a = x,a satisfying x a ;b = 0 and
3
also x a x a = 1. From the Ricci Identity R a bcd x d = 0. It follows from [2] that affine
vector fields in this case are
X = (c1 x + c 2 )
∂ +Y ∂x
(8)
where c1 , c 2 ∈ ℜ and Y is a homothetic vector field in the induced geometry on each of the three dimensional submanifolds of constant x. Next suppose M has type R7 . Then each p ∈ M has a neighborhood U which decomposes metrically as U = U 1 × U 2 , where U 1 is a two dimensional submanifold of U with an induced metric of Lorentz signature and U 2 is a two dimensional submanifold of U with positive definite induced metric. The spacetime metric on U is given by [1] ds 2 = PAB dx A dx B + Qαβ dx α dx β
(9)
where PAB = PAB ( x C ),∀ A, B,C = 0,1 and Qαβ = Qαβ ( x γ ),∀α , β ,γ = 2, 3 and the above space-time is clearly 2+2 decomposable. The space-time (8) admits two recurrent vector fields [5] l and n i.e. l a;b = l a pb and na ;b = na pb where pb is the recurrent 1-form. It also admits two covariantly constant second order symmetric tensors which are 2l ( a nb ) and ( x a xb + y a y b ). The rank of the 6 × 6 Riemann matrix is two. It follows from [2] that if X is an affine vector field on M then
X decomposes as X = X1 + X 2
(10)
where the vector fields X 1 and X 2 are tangent to the two dimensional timelike and spacelike submanifolds, respectively. It also follows from [2] that X 1 and X 2 are homothetic vector fields in their respective submanifolds with their
induced geometry. Conversely, every pair of affine vector fields, one in the timelike submanifolds and one spacelike submanifolds give rise to a affine vector field in space-time. Now suppose that M has type R4 . Then each p ∈ M has a neighborhood U which decomposes metrically as U = U 1 × U 2 × U 3 , where U 1 and U 2 are one
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dimensional submanifold of U and U 3 is a two dimensional submanifold of U . The space-time metric on U is given by [2]
ds 2 = − dt 2 + dx 2 + g AB dx A dx B
( A, B = 2, 3)
(11)
(C = 2, 3). The above space-time is clearly
where g AB depends only on x C
1+1+2 decomposable. The curvature rank of the above space-time is one and there exist two independent nowhere zero unit timelike and spacelike covariantly constant vector field t a = t ,a and x a = x,a satisfying t a ;b = 0 and x a ;b = 0 . From the Ricci identity R a bcd t a = 0 and R a bcd x a = 0. It follows from [2] that affine vector fields in this case are
X = (c1t + c 2 x + c3 )
∂ ∂ + (c 4 t + c 5 x + c 6 ) + Y ∂t ∂x
(12)
where c1 , c 2 , c3 , c 4 , c5 , c6 ∈ ℜ and Y is a homothetic vector on each of two dimensional submanifolds of constant t and x. Now suppose that M
has type R2 . Here each p ∈ M
admits a
neighborhood U which decomposes metrically as U = U 1 × U 2 × U 3 , where U 1 and U 2 are one dimensional submanifold of U and U 3 is a two dimensional submanifold of U . The space-time metric on U is given by [2]
ds 2 = dy 2 + dz 2 + g AB dx A dx B where g AB depends only on x C
( A, B = 0,1)
(13)
(C = 0, 1). The above space-time is clearly
1+1+2 decomposable. The curvature rank of the above space-time is one and there exist two independent nowhere zero unit timelike and spacelike covariantly constant vector field y a = y ,a and z a = z ,a satisfying y a ;b = 0 and z a;b = 0. From the Ricci identity R a bcd y a = 0 and R a bcd z a = 0. It follows from [2] that affine vector fields in this case are
X = (c1 y + c 2 z + c3 )
∂ ∂ + (c 4 y + c 5 z + c 6 ) + Y ∂y ∂z
(14)
where c1 , c 2 , c3 , c 4 , c5 , c6 ∈ ℜ and Y is a homothetic vector on each of two dimensional submanifolds of constant y and z.
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3. MAIN RESULTS
As mentioned in section 2, the space-times which can admit proper affine vector fields having holonomy type R2 , R3 , R4 , R6 , R7 , R8 , R10 , R11 or R13 . It also follows from [5] that the rank of the 6 × 6 Riemann matrix is atmost three. Here in this paper we will consider the rank of the 6 × 6 Riemann matrix to study affine vector fields in plane symmetric static space-time. Consider plane symmetric static space-time in the usual coordinate system (t , x, y, z ) with line elememt [7]
ds 2 = −e v ( x ) dt 2 + dx 2 + e µ ( x ) (dy 2 + dz 2 )
(15)
It follows from [7], the above space-time admits four independent Killing vector fields which are ∂ , ∂t
∂ , ∂y
y
∂ ∂ −z , ∂y ∂z
∂ . ∂z
The non-zero independent components of the Riemann tensor are 1 (2v ′′ + v ′ 2 )e −v ≡ α 1 , 4 1 R 02 02 = R 03 03 = − µ ′ v ′ ≡ α 2 , 4 1 R 1212 = R 1313 = − (2 µ ′′ + µ ′ 2 ) ≡ α 3 , 4 1 R 23 23 = − µ ′ 2 ≡ α 4 . 4 R 01 01 =
(16)
Writing the curvature tensor with components R ab cd at p as a 6X6 symmetric matrix in a well known way [6]
R ab cd = diag (α 1 , α 2 , α 2 , α 3 , α 3 , α 4 ) where α 1 , α 2 , α 3 and α 4 are real functions of x. The 6-dimensional labeling is in the order 01, 02, 03, 12, 13, 23 with x 0 = t. We are only interested in those case when the rank of the 6 × 6 Riemann matrix is less than or equal to three (excluding flat cases). We thus obtain the following cases: (A) Rank ≤ 3, when v ∈ R, µ = µ (r ), (B) Rank =1, when v = v( x), λ ∈ R,
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(C) Rank =3, when v = v( x), µ = µ ( x), 2v ′′ + v ′ 2 = 2µ ′′ + µ ′ 2 = 0. We consider each case in turn.
Case A In this case v ∈ R, µ = µ (r ). First suppose that the rank of the 6 × 6 Riemann matrix is 3 and there exists a unique (up to a multiple) nowhere zero timelike vector field t a = t ,a such that t a ;b = 0 (and so, from the Ricci idenity
R a bcd t a = 0 ). The line element can, after a recaling of t, be written in the form ds 2 = −dt 2 + (dx 2 + e µ ( x ) (dy 2 + dz 2 )).
(17)
The space-time is clearly 1+3 decomposable. The affine vector fields in this case [2] are
X = (c8 t + c 9 )
∂ + X′ ∂t
(18)
where c8 , c9 ∈ R and X ′ is a homothetic vector field in the induced geometry on each of the three dimensional submanifolds of constant t. The completion of case A necessities finding an homothetic vector fields in the induced geometry of the submanifolds of constant t. The induced metric g αβ (where α , β = 1, 2, 3) with nonzero components is given by
g11 = 1, g 22 = e µ ( x ) , g 33 = e µ ( x ) .
(19)
A vector field X ′ is called homothetic vector field if it satisfies L X ' g αβ = 2cgαβ , where c ∈ R. One can expand by using (19) to get
X 1,1 = c
(20)
X 1, 2 + e µ ( x ) X 2 ,1 = 0
(21)
X 1,3 + e µ ( x ) X 3 ,1 = 0
(22)
µ ′ X 1 + 2 X 2 , 2 = 2c
(23)
X 2 ,3 + X 3 , 2 = 0
(24)
µ ′ X 1 + X 3 ,3 = 2c.
(25)
From equations (20), (21) and (22) we get
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X 1 = cx + A1 ( y, z ), X 2 = − A1 y ( y, z ) ∫ e − µ dx + A 2 ( y, z )
X 3 = − A1 z ( y, z ) ∫ e − µ dx + A 3 ( y, z )
(26)
where A1 ( y, z ) A 2 ( y, z ) and A 3 ( y, z ) are functions of integration. Now consider equation (24) and differentiate with respect to x and using (26) one finds A1 yz ( y, z ) = 0 ⇒ A 1 ( y, z ) = B1 ( y ) + B 2 ( z )
where
B1 ( y)
and
B 2 ( y)
are
functions of integration. Substituting back into (26) we get X 1 = cx + B 1 ( y ) + B 2 ( z ), X 2 = − B 1 y ( y ) ∫ e − µ dx + A 2 ( y, z ), X 3 = − B 2 z ( z ) ∫ e − µ dx + A 3 ( y, z ).
(27)
Substituting equation (25) from (23) and differentiate with respect to x and using (27) gives
B 1 yy ( y ) − B 2 zz ( z ) = 0,
B 1 yyy ( y ) = 0 ⇒ B 1 ( y ) = B 2 ( z) =
differentiate with respect to y gives
c1 2 y + c 2 y + c3 . 2
Substituting
back
we
get
c1 2 z + c 4 z + c5 where c1 ,c 2 ,c3 ,c 4 ,c5 ∈ ℜ and so equation (27) becomes 2 X 1 = cx + (
c1 2 c y + c 2 y + c3 ) + ( 1 z 2 + c 4 z + c5 ), 2 2
X 2 = −(c1 y + c 2 ) ∫ e − µ dx + A 2 ( y, z ), X 3 = −(c1 z + c 4 ) ∫ e − µ dx + A 3 ( y, z ).
(28)
Now consider equation (23), taking partial derivatives with respect to y and using (28) to get µ ′(c1 y + c 2 ) + 2 A 2 yy ( y, z ) = 0. Differentiating with respect to x gives
µ ′′(c1 y + c 2 ) = 0 and there now exist two possibilities: (1) µ ′′ = 0, (c1 y + c 2 ) ≠ 0,
(2) µ ′′ ≠ 0, (c1 y + c 2 ) = 0.
Class A1
µ ′′ = 0 ⇒ µ = ax + b, where a,b ∈ R (a ≠ 0) and (c1 y + c 2 ) ≠ 0. Using this information in (28) and solving equations (20) to (25), one finds that homothetic vector fields in this case are Killing vector fields which are given by X 1 = c 2 y + c 4 z + c5 ,
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X 2 = c2
1 − ax −b a a e + c 2 ( z 2 − y 2 ) − (c3 yz + c 4 y ) + c5 z + c6 , a 4 2
X 3 = c2
1 −ax −b a a e + c3 (− z 2 + y 2 ) − (c 2 yz + c 4 y ) − c5 z + c7 a 4 2
(29)
where c 2 ,c3 ,c 4 ,c5 ,c6 , c7 ∈ R. Thus the submanifolds of constant t are of constant curvature and affine vector fields in this case are X 0 = c8 t + c 9 ,
X 1 = c 2 y + c 4 z + c5 ,
X 2 = c2
1 − ax −b a a e + c 2 ( z 2 − y 2 ) − (c3 yz + c 4 y ) + c5 z + c6 , a 4 2
X 3 = c2
1 − ax −b a a e + c3 (− z 2 + y 2 ) − (c 2 yz + c 4 z ) − c5 y + c7 . a 4 2
(30)
Case A2 In this case µ ′′ ≠ 0 and (c1 y + c 2 ) = 0 ⇒ c1 = c 2 = 0 and A 2 yy ( y, z ) = 0 or A 2 ( y, z ) = yB 3 ( z ) + B 4 ( z ) where B 3 ( z ) and B 4 ( z ) are functions of integrations. Using the information in (28) and solving equations (20) to (25), one finds that there exist the following two possible subcasses: (i)
2 µ ′′ + µ ′ 2 ≠ 0
(ii)
2 µ ′′ + µ ′ 2 = 0
Case A2i In this case 2 µ ′′ + µ ′ 2 ≠ 0 and that the rank of the 6 × 6 Riemann matrix is 3. Using the above information in equations (20) to (25) and solving them, one finds that homothetic vector fields in this case are Killing vector fields which are given by X 1 = 0,
X 2 = −c1 z + c3 , X 3 = c1 y + c 2 .
(31)
where c1 ,c 2 ,c3 ∈ R and affine vector fields in this case are (from equation (18)) X 0 = c8 t + c9 , X 1 = 0, X 3 = c1 y + c 2 .
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X 2 = −c1 z + c3 ,
Case A2ii In this case 2 µ ′′ + µ ′ 2 = 0 ⇒ µ = ln(ax + b) 2 , where a,b ∈ R (a ≠ 0). Here the rank of the 6 × 6 Riemann matrix is one and there exist two independent nowhere zero solutions of equation (4) i.e. R a bcd t a = R a bcd x a = 0 where x a = x,a and t a = t ,a are the spacelike and timelike vector field, respectively. The sapcetime (32) admits only one independent nowhere zero timelike covariantly constant vector field t a satisfying t a ;b = 0. After rescaling of t the line element takes the form ds 2 = dt 2 + dx 2 + (ax + b) 2 (dy 2 + dz 2 ).
(32)
The above space-time is clearly 1+3 decomposable but the rank of 6X6 Riemann matrix is one. Substituting the above information into affine equations one finds, affine vector fields in this case X 0 = c 4 t + c5 x + c 6 ,
X 1 = c 7 t + c8 x,
X 2 = −c1 z + c3 ,
X 3 = c1 y + c 2
(33)
where c1 , c 2 , c3 , c 4 , c5 , c6 , c7 , c8 ∈ R. This completes case (A).
Case B In this case µ ∈ R, v = v( x) and the rank of the 6 × 6 Riemann matrix is one and there exist two independent nowhere zero spacelike vector field y a = y,a and
z a = z ,a
satisfying
y a ;b = 0 and
z a;b = 0. From the Ricci identity
R a bcd y a = R a bcd z a = 0. The line element can, after a rescaling of y and z, be written as ds 2 = dy 2 + dz 2 + (−e v ( x ) dt 2 + dx 2 ).
(34)
Clearly the above space-time is 1+1+2 decomposable. The affine vector fields in this case are [2]
X = (c1 y + c 2 z + c3 )
∂ ∂ + (c 4 y + c5 z + c 6 ) + X ′ ∂y ∂z
10
(35)
where c1 , c 2 , c3 , c 4 , c5 , c6 ∈ R and X ′ is a homothetic vector fields in each of two dimensional submanifolds of constant y and z. The next step is to finding homothetic vector fields in the induced geometry of the submanifolds of constant
y and z. The induced metric g AB (where A, B = 0,1) with nonzero components is given by
g 00 = −e v ( x ) , g11 = 1.
(36)
A vector field X ′ is called homothetic vector field if it satisfies L X ' g AB = 2cg AB , where c ∈ R. One can expand by using (36) to get
v ′X 1 + 2 X 0 , 0 = 2c
(37)
X 1,0 − e v X 0 ,1 = 0
(38)
X 1,1 = c.
(39)
Equation (39) gives X 1 = cx + A1 (t ), where A1 (t ) is a function of integration. Using value of X 1 in equation (38) we get X 0 = At1 (t ) ∫ e − v dx + A 2 (t ), where
A 2 (t ) is a function of integration. If one proceeds further, after a straightforward calculation one finds that proper homothetic vector field exist if and only if
v = ln(ax 2 ), where a ∈ R − {0}. Substituting the value of v into (16), one finds that the rank of 6 × 6 Riemann matrix reduces to zero thus giving a contradiction (since we are assuming that the rank of 6 × 6 Riemann matrix is one). So homothetic vector fields in the induced geometry of constant y and z are Killing vector fields. If one proceeds further one finds there exist two possibilities: (1) v ′′e v = c
(2) v ′′e v ≠ c
where c ∈ R.
Case B1 In this case further three possibilities exist (i) c > 0,
(ii) c < 0,
We will consider each case in turn. (i)
Affine vector fields in this case are
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(iii) c = 0.
X0 =
c c c (c7 cos t − c8 sin t ) ∫ e −v dx + c9 , 2 2 2
c c t + c8 cos t 2 2 X 2 = c1 y + c 2 z + c3 , X 3 = c 4 y + c5 z + c6 . X 1 = c7 sin
(40)
provided that v ′′e v = c. Where c7 , c8 , c9 ∈ R. (ii)
In this case c < 0. Put c = − N , where N ∈ R ( N > 0). Affine vector fields
in this case are
X0 =
N N N (c7 cosh t + c8 sinh t ) ∫ e −v dx + c9 , 2 2 2
N N t + c8 cosh t 2 2 X 2 = c1 y + c 2 z + c3 , X 3 = c 4 y + c5 z + c6 . X 1 = c7 sinh
(41)
provided that v ′′e v = − N . Where c7 , c8 , c9 ∈ R. (iii)
In this case c = 0 ⇒ v = ax + b, where a, b ∈ R(a ≠ 0). Affine vector
fields in this case are
c7 −( ax +b ) + c9 e X 1 = c 7 t + c8 a 2 X = c1 y + c 2 z + c3 , X 3 = c 4 y + c5 z + c6 . X 0 =−
(42)
where c7 , c8 , c9 ∈ R.
Case B2 Affine vector fields in this case are X 0 = c7 ,
X 1 = 0,
X 2 = c1 y + c 2 z + c3 ,
X 3 = c 4 y + c5 z + c 6 .
(43)
where c1 ∈ R. This completes case B.
Case C In this case v = v(r ), µ = µ ( x), 2v ′′ + v ′ 2 = 0 and 2µ ′′ + µ ′ 2 = 0. The equations
2µ ′′ + µ ′ 2 = 0
and
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a 2v ′′ + v ′ 2 = 0 ⇒ v = ln( x + b) 2 2
and
c 2µ ′′ + µ ′ 2 = 0 ⇒ µ = ln( x + d ) 2 where a, b, c, d ∈ R (a, c ≠ 0) and a ≠ c, b ≠ d 2 (by assumption). The rank of the 6 × 6 Riemann matrix is 3 and there exist a unique (up to a multiple) nowhere zero spacelike vector field x a = x,a such that
R a bcd x a = 0 and x a ;b ≠ 0. The line element is a c ds 2 = −( x + b) 2 dt 2 + dx 2 + ( x + d ) 2 (dy 2 + dz 2 ) 2 2
(44)
Substituting the above information into affine equations and after a strightforward calculation one find affine vector fields in this case are
X 0 = c2 ,
X 1 = 0,
X 2 = c3 z + c 4 ,
X 3 = −c3 y + c5
(45)
where c 2 , c3 , c 4 , c5 ∈ ℜ. Clearly affine vector fields in this case are Killing vector fields. Now consider the case when a = c ≠ 0, b = d , the line element is
a a ds 2 = −( x + b) 2 dt 2 + dx 2 + ( x + b) 2 (dy 2 + dz 2 ). 2 2
(46)
It follows from [9] affine vector fields in this case are
X 0 = c1 y + c 2 z + c3 , X 1 = c7 x, X 2 = c1t − c 4 z + c5 , X 3 = c 2 t + c 4 z + c6
(47)
Where c1 , c 2 , c3 , c 4 , c5 , c6 , c7 ∈ R.
SUMMARY In this paper a study of plane symmetric static space-times according to their proper affine vector fields is given. An approach is developed to study proper affine vector fields in the above space-times by using the rank of the 6X6 Riemann matrix, holonomy and direct integration techniques. From the above study we obtain the following results: (i)
The case when the rank of the 6X6 Riemann matrix is three and there
exists a nowhere zero independent spacelike vector field which is the solution of
13
equation (4) and is not covariantly constant. This is the space-time (44) and it admits affine vector fields which are Killing vector fields (see for details Case C). (ii)
The case when the rank of the 6X6 Riemann matrix is three and there exist
a nowhere zero independent spacelike vector field which is a solution of equation (4) and is not covariantly constant. This is the space-time (46) and it admits the proper affine vector fields (equation 47). (iii)
The case when the rank of the 6X6 Riemann matrix one there exist two
nowhere zero independent solutions of equation (4) but only one independent nowhere zero covariantly constant vector field. This is the space-time (32) and it admits the proper affine vector fields (see Case A2ii). (iv)
The case when the rank of the 6X6 Riemann matrix is two or three and
there exists a nowhere zero independent timelike vector field which is the solution of equation (4) and also covariantly constant. This is the space-time (17) and it admits the proper affine vector fields (see Cases A1 and A2i). (v)
The case when the rank of the 6X6 Riemann matrix one there exist two
nowhere zero independent spacelike vector fields which are solutions of equation (4) and which are covariantly constant vector field. This is the space-time (34) and it admits the proper affine vector fields (see Case B1i, B1ii, B1iii and B2).
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